Functional harmony in rank-2 temperaments: Difference between revisions

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== Role swapping vs breaking out of mos ==

~~{{Todo|inline=1|expand|comment=Discuss the non-swapping technique in each example. }}~~

A greater problem arises when the temperament in question is not generated by a [[3/2|perfect fifth (3/2)]]. In septimal meantone, the most [[consonance and dissonance|consonant]] [[interval]] besides the [[octave]], the perfect fifth, is a [[generator]], so that the most consonant interval is also the most abundant interval in any [[mos scale]]. This is not true for any temperament that is not generated by a perfect fifth. In these temperaments, we have to choose what our dominant and subdominant functions are based on.

Here we discuss the technique where "dominant" and "subdominant" are no longer related to tonic by the fifth, but by the generator. This can be called ''role swapping'' as we are essentially swapping the functional roles of harmonic 3 and a higher harmonic according to their temperamental complexities (number of generator steps). Since mos scales are created from stacking generators, this leads to an approach closely adherent to mos scales, and depending on the specific scale pattern, it can sound very xenharmonic.

In contrast to role swapping, ~~there is a~~ one-for-all solution to harmony in temperaments regardless of how far the primes are on the generator chain~~. It~~ entails using the same old chords as the basis of harmony, and keeping the roles of each prime as in meantone. This means dominant is always a perfect fifth over tonic, for example. A consequence is we must step out of the logic of mos scales, as they are often too restrictive without the many fifths to stack. Try starting unlimited and thinking directly about [[ratio]]s and [[comma pump]]s. You might end up with a diatonic-like scale with comma-level inflections here and there, but it is also possible to slap whatever scale you like, using inflections to reach the ratios.

In contrast to role swapping, the one-for-all solution to harmony in temperaments regardless of how far the primes are on the generator chain entails using the same old chords as the basis of harmony, and keeping the roles of each prime as in meantone. This means dominant is always a perfect fifth over tonic, for example. A consequence is we must step out of the logic of mos scales, as they are often too restrictive without the many fifths to stack. Try starting unlimited and thinking directly about [[ratio]]s and [[comma pump]]s. You might end up with a diatonic-like scale with comma-level inflections here and there, but it is also possible to slap whatever scale you like, using inflections to reach the ratios.

=== Example: magic ===

In terms of generator steps of [[magic]], the just major triad (1–5/4–3/2) and just minor triad (1–6/5–3/2) are 0–1–5 and 0–4–5; this is similar, but also in clear contrast to the 0–4–1 and 0–(−3)–1 of meantone. These chords can be used as the basis of harmony, with the roles of 3 and 5 swapped according to their temperamental complexities. Thus a "dominant" chord is 5/4 over tonic; a "subdominant" chord is 5/4 under tonic. The 7-tone mos contains a tonic and a "dominant" chord. The 10-tone mos is good for encapsulating tonic, "pre-dominant", and "dominant" functions.

In terms of generator steps of [[magic]], the just major triad (1–5/4–3/2) and just minor triad (1–6/5–3/2) are 0–1–5 and 0–4–5; this is similar, but also in clear contrast to the 0–4–1 and 0–(−3)–1 of meantone. These chords can be used as the basis of harmony, with the roles of 3 and 5 swapped according to their temperamental complexities. Thus a "dominant" chord is 5/4 over tonic; a "subdominant" chord is 5/4 under tonic. The 7-tone mos contains a tonic and a "dominant" chord. The 10-tone mos is good for encapsulating tonic, "pre-dominant", and "dominant" functions. The suspended chord of meantone is made of two generators stacked, and doing the same in magic, we also have the augmented triad (1–5/4–14/9) as the magic analog of the suspended chord of meantone.

~~The suspended chord of meantone is made~~ of ~~two generators stacked. Doing the same in magic~~, we ~~also have~~ the ~~augmented~~ triad (~~1–5/4–14/9~~) as the ~~magic analog~~ of the ~~suspended chord of meantone~~.

Instead of thinking about mos scales, we can keep the tonic, dominant, and subdominant functions related by fifths. The classical major triad on tonic is 0–1–5; on dominant, 5–6–10; and on subdominant, (-5)–(-4)–0. This is essentially working in a rank-3 space (or 2-dimensional [[lattice]] up to octave equivalence) but the edges are wrapped around due to the equivalence from the commas tempered out. The only thing that distinguishes it from JI is that you should in some way use the commas tempered out to prove the worth of the intonational compromises; otherwise you could simply choose JI.

=== Example: hanson ===

In terms of generator steps of [[hanson]], the just major triad (1–5/4–3/2) and just minor triad (1–6/5–3/2) are 0–5–6 and 0–1–6; like in magic, these chords can be used as the basis of harmony, but the intervals whose roles are being swapped here are 3 and 3/5. A "dominant" chord is 6/5 over tonic; a "subdominant" chord is 6/5 under tonic. The 11-tone mos might be a good place to start since the 7-tone mos barely contains anything more a tonic chord.

If instead we keep the tonic, dominant, and subdominant functions related by fifths, then the classical major triad on tonic is 0–5–6; on dominant, 6–11–12; and on subdominant, (-6)–(-1)–0.

=== Example: orwell ===

[[Orwell]] is generated by ~7/6, so ~~the most basic form of harmony~~ could be 1–7/6–3/2 (0–1–7) and 1–9/7–3/2 (0–6–7), or 1–7/4–3 (0–8–7) and 1–12/7–3 (0–(-1)–7), or tetrads such as 1–7/6–3/2–7/4 (0–1–7–8). The intervals whose roles are being swapped here are 3 and 5 and then 5 and 7. A "dominant" chord is either 7/6 or 12/7 over tonic; a "subdominant" chord is either 7/6 or 12/7 under tonic. The 9-tone mos contains a tonic and a "dominant" triad. The 13-tone mos is good for encapsulating tonic, "pre-dominant", and "dominant" functions, triads and ~~tetrads alike~~.

[[Orwell]] is generated by ~7/6, so chords could be built around it as 1–7/6–3/2 (0–1–7) and 1–9/7–3/2 (0–6–7), or 1–7/4–3 (0–8–7) and 1–12/7–3 (0–(-1)–7), or tetrads such as 1–7/6–3/2–7/4 (0–1–7–8). The intervals whose roles are being swapped here are 3 and 5 and then 5 and 7. Since two generator steps give ~11/8, 1–11/8–3/2–7/4 (0–2–7–8) is also a characteristic otonal tetrad of orwell. To 1–7/6–3/2–7/4 we may add 11/8, or to 1–11/8–3/2–7/4 we may add 7/6, to form an essentially tempered pentad, 1–7/6–11/8–3/2–7/4 (0–1–2–7–8). Its inverse is 1–12/11–9/7–3/2–12/7 (0–5–6–7–(-1)), which can serve as a minor counterpart. A "dominant" chord is either 7/6 or 12/7 over tonic; a "subdominant" chord is either 7/6 or 12/7 under tonic. The 9-tone mos contains a tonic and a "dominant" triad. The 13-tone mos is good for encapsulating tonic, "pre-dominant", and "dominant" functions, triads to pentads alike.

If instead we keep the tonic, dominant, and subdominant functions related by fifths, then the above pentad on tonic is 0–1–2–7–8; on dominant, 7–8–9–14–15; and on subdominant, (-7)–(-6)–(-5)–0–1.

=== Example: sensi ===

[[Sensi]] is generated by ~9/7, so ~~the most basic form of harmony~~ could be 1–9/7–3/2 (0–1–7) and 1–7/6–3/2 (0–6–7). The intervals whose roles are being swapped here are 3 and 5 and then 5 and 9/7. A "dominant" chord is 9/7 over tonic; a "subdominant" chord is 9/7 under tonic. The 11-tone mos might be a good place to start since the 8-tone mos barely contains anything more than a tonic chord.

[[Sensi]] is generated by ~9/7, so chords could be built around it as 1–9/7–3/2 (0–1–7) and 1–7/6–3/2 (0–6–7). The intervals whose roles are being swapped here are 3 and 5 and then 5 and 9/7. Since two generator steps give ~5/3, 1–7/6–3/2–5/3 (0–6–7–2) is also a characteristic otonal tetrad of sensi; its inverse is 1–9/7–3/2–9/5 (0–1–7–5). In addition, 1–9/7–3/2–5/3 (0–1–7–2) is a very characteristic essentially tempered tetrad; its inverse is 1–7/6–3/2–9/5 (0–6–7–5). A "dominant" chord is 9/7 over tonic; a "subdominant" chord is 9/7 under tonic. The 11-tone mos might be a good place to start since the 8-tone mos barely contains anything more than a tonic chord.

If instead we keep the tonic, dominant, and subdominant functions related by fifths, then the above tetrads on tonic are 0–6–7–2 and 0–1–7–2; on dominant, 7–13–14–9 and 7–8–14–9; and on subdominant, (-7)–(-1)–0–(-5) and (–7)–(-6)–0–(-5).

=== Example: würschmidt ===

[[Würschmidt]] is just about on the complexity level where role swapping stops to work. The 10-tone mos contains a tonic and a "dominant" chord. For full analog of traditional harmony you need an unwieldy 13-tone mos, with more notes than the traditional chromatic scale and nearly twice the number of notes as the diatonic scale.

~~Instead of thinking about~~ the ~~mos scheme~~, ~~we can pick our desired intervals however we like. If you want~~ the classical major triad on tonic~~, use~~ 0–1–8; on dominant, ~~use~~ 8–9–16; and on subdominant, ~~use~~ (-8)–(-7)–0. The possibilities are theoretically infinite. This is essentially working in a rank-3 space (or 2-dimensional [[lattice]] up to octave equivalence) but the edges are wrapped around due to the equivalence from the commas tempered out. The only thing that distinguishes it from JI is that you should in some way use the commas tempered out to prove the worth of the intonational compromises; otherwise you could simply choose JI.

If instead we keep the tonic, dominant, and subdominant functions related by fifths, then the classical major triad on tonic is 0–1–8; on dominant, 8–9–16; and on subdominant, (-8)–(-7)–0.

== Generalization ==

@@ Line 28: / Line 28: @@
 == Role swapping vs breaking out of mos ==
-{{Todo|inline=1|expand|comment=Discuss the non-swapping technique in each example. }}
 A greater problem arises when the temperament in question is not generated by a [[3/2|perfect fifth (3/2)]]. In septimal meantone, the most [[consonance and dissonance|consonant]] [[interval]] besides the [[octave]], the perfect fifth, is a [[generator]], so that the most consonant interval is also the most abundant interval in any [[mos scale]]. This is not true for any temperament that is not generated by a perfect fifth. In these temperaments, we have to choose what our dominant and subdominant functions are based on.
 Here we discuss the technique where "dominant" and "subdominant" are no longer related to tonic by the fifth, but by the generator. This can be called ''role swapping'' as we are essentially swapping the functional roles of harmonic 3 and a higher harmonic according to their temperamental complexities (number of generator steps). Since mos scales are created from stacking generators, this leads to an approach closely adherent to mos scales, and depending on the specific scale pattern, it can sound very xenharmonic.
-In contrast to role swapping, there is a one-for-all solution to harmony in temperaments regardless of how far the primes are on the generator chain. It entails using the same old chords as the basis of harmony, and keeping the roles of each prime as in meantone. This means dominant is always a perfect fifth over tonic, for example. A consequence is we must step out of the logic of mos scales, as they are often too restrictive without the many fifths to stack. Try starting unlimited and thinking directly about [[ratio]]s and [[comma pump]]s. You might end up with a diatonic-like scale with comma-level inflections here and there, but it is also possible to slap whatever scale you like, using inflections to reach the ratios.
+In contrast to role swapping, the one-for-all solution to harmony in temperaments regardless of how far the primes are on the generator chain entails using the same old chords as the basis of harmony, and keeping the roles of each prime as in meantone. This means dominant is always a perfect fifth over tonic, for example. A consequence is we must step out of the logic of mos scales, as they are often too restrictive without the many fifths to stack. Try starting unlimited and thinking directly about [[ratio]]s and [[comma pump]]s. You might end up with a diatonic-like scale with comma-level inflections here and there, but it is also possible to slap whatever scale you like, using inflections to reach the ratios.
 === Example: magic ===
-In terms of generator steps of [[magic]], the just major triad (1–5/4–3/2) and just minor triad (1–6/5–3/2) are 0–1–5 and 0–4–5; this is similar, but also in clear contrast to the 0–4–1 and 0–(−3)–1 of meantone. These chords can be used as the basis of harmony, with the roles of 3 and 5 swapped according to their temperamental complexities. Thus a "dominant" chord is 5/4 over tonic; a "subdominant" chord is 5/4 under tonic. The 7-tone mos contains a tonic and a "dominant" chord. The 10-tone mos is good for encapsulating tonic, "pre-dominant", and "dominant" functions.
+In terms of generator steps of [[magic]], the just major triad (1–5/4–3/2) and just minor triad (1–6/5–3/2) are 0–1–5 and 0–4–5; this is similar, but also in clear contrast to the 0–4–1 and 0–(−3)–1 of meantone. These chords can be used as the basis of harmony, with the roles of 3 and 5 swapped according to their temperamental complexities. Thus a "dominant" chord is 5/4 over tonic; a "subdominant" chord is 5/4 under tonic. The 7-tone mos contains a tonic and a "dominant" chord. The 10-tone mos is good for encapsulating tonic, "pre-dominant", and "dominant" functions. The suspended chord of meantone is made of two generators stacked, and doing the same in magic, we also have the augmented triad (1–5/4–14/9) as the magic analog of the suspended chord of meantone.
-The suspended chord of meantone is made of two generators stacked. Doing the same in magic, we also have the augmented triad (1–5/4–14/9) as the magic analog of the suspended chord of meantone.
+Instead of thinking about mos scales, we can keep the tonic, dominant, and subdominant functions related by fifths. The classical major triad on tonic is 0–1–5; on dominant, 5–6–10; and on subdominant, (-5)–(-4)–0. This is essentially working in a rank-3 space (or 2-dimensional [[lattice]] up to octave equivalence) but the edges are wrapped around due to the equivalence from the commas tempered out. The only thing that distinguishes it from JI is that you should in some way use the commas tempered out to prove the worth of the intonational compromises; otherwise you could simply choose JI.
 === Example: hanson ===
 In terms of generator steps of [[hanson]], the just major triad (1–5/4–3/2) and just minor triad (1–6/5–3/2) are 0–5–6 and 0–1–6; like in magic, these chords can be used as the basis of harmony, but the intervals whose roles are being swapped here are 3 and 3/5. A "dominant" chord is 6/5 over tonic; a "subdominant" chord is 6/5 under tonic. The 11-tone mos might be a good place to start since the 7-tone mos barely contains anything more a tonic chord.
+If instead we keep the tonic, dominant, and subdominant functions related by fifths, then the classical major triad on tonic is 0–5–6; on dominant, 6–11–12; and on subdominant, (-6)–(-1)–0.
 === Example: orwell ===
-[[Orwell]] is generated by ~7/6, so the most basic form of harmony could be 1–7/6–3/2 (0–1–7) and 1–9/7–3/2 (0–6–7), or 1–7/4–3 (0–8–7) and 1–12/7–3 (0–(-1)–7), or tetrads such as 1–7/6–3/2–7/4 (0–1–7–8). The intervals whose roles are being swapped here are 3 and 5 and then 5 and 7. A "dominant" chord is either 7/6 or 12/7 over tonic; a "subdominant" chord is either 7/6 or 12/7 under tonic. The 9-tone mos contains a tonic and a "dominant" triad. The 13-tone mos is good for encapsulating tonic, "pre-dominant", and "dominant" functions, triads and tetrads alike.
+[[Orwell]] is generated by ~7/6, so chords could be built around it as 1–7/6–3/2 (0–1–7) and 1–9/7–3/2 (0–6–7), or 1–7/4–3 (0–8–7) and 1–12/7–3 (0–(-1)–7), or tetrads such as 1–7/6–3/2–7/4 (0–1–7–8). The intervals whose roles are being swapped here are 3 and 5 and then 5 and 7. Since two generator steps give ~11/8, 1–11/8–3/2–7/4 (0–2–7–8) is also a characteristic otonal tetrad of orwell. To 1–7/6–3/2–7/4 we may add 11/8, or to 1–11/8–3/2–7/4 we may add 7/6, to form an essentially tempered pentad, 1–7/6–11/8–3/2–7/4 (0–1–2–7–8). Its inverse is 1–12/11–9/7–3/2–12/7 (0–5–6–7–(-1)), which can serve as a minor counterpart. A "dominant" chord is either 7/6 or 12/7 over tonic; a "subdominant" chord is either 7/6 or 12/7 under tonic. The 9-tone mos contains a tonic and a "dominant" triad. The 13-tone mos is good for encapsulating tonic, "pre-dominant", and "dominant" functions, triads to pentads alike.
+If instead we keep the tonic, dominant, and subdominant functions related by fifths, then the above pentad on tonic is 0–1–2–7–8; on dominant, 7–8–9–14–15; and on subdominant, (-7)–(-6)–(-5)–0–1.
 === Example: sensi ===
-[[Sensi]] is generated by ~9/7, so the most basic form of harmony could be 1–9/7–3/2 (0–1–7) and 1–7/6–3/2 (0–6–7). The intervals whose roles are being swapped here are 3 and 5 and then 5 and 9/7. A "dominant" chord is 9/7 over tonic; a "subdominant" chord is 9/7 under tonic. The 11-tone mos might be a good place to start since the 8-tone mos barely contains anything more than a tonic chord.
+[[Sensi]] is generated by ~9/7, so chords could be built around it as 1–9/7–3/2 (0–1–7) and 1–7/6–3/2 (0–6–7). The intervals whose roles are being swapped here are 3 and 5 and then 5 and 9/7. Since two generator steps give ~5/3, 1–7/6–3/2–5/3 (0–6–7–2) is also a characteristic otonal tetrad of sensi; its inverse is 1–9/7–3/2–9/5 (0–1–7–5). In addition, 1–9/7–3/2–5/3 (0–1–7–2) is a very characteristic essentially tempered tetrad; its inverse is 1–7/6–3/2–9/5 (0–6–7–5). A "dominant" chord is 9/7 over tonic; a "subdominant" chord is 9/7 under tonic. The 11-tone mos might be a good place to start since the 8-tone mos barely contains anything more than a tonic chord.
+If instead we keep the tonic, dominant, and subdominant functions related by fifths, then the above tetrads on tonic are 0–6–7–2 and 0–1–7–2; on dominant, 7–13–14–9 and 7–8–14–9; and on subdominant, (-7)–(-1)–0–(-5) and (–7)–(-6)–0–(-5).
 === Example: würschmidt ===
 [[Würschmidt]] is just about on the complexity level where role swapping stops to work. The 10-tone mos contains a tonic and a "dominant" chord. For full analog of traditional harmony you need an unwieldy 13-tone mos, with more notes than the traditional chromatic scale and nearly twice the number of notes as the diatonic scale.
-Instead of thinking about the mos scheme, we can pick our desired intervals however we like. If you want the classical major triad on tonic, use 0–1–8; on dominant, use 8–9–16; and on subdominant, use (-8)–(-7)–0. The possibilities are theoretically infinite. This is essentially working in a rank-3 space (or 2-dimensional [[lattice]] up to octave equivalence) but the edges are wrapped around due to the equivalence from the commas tempered out. The only thing that distinguishes it from JI is that you should in some way use the commas tempered out to prove the worth of the intonational compromises; otherwise you could simply choose JI.
+If instead we keep the tonic, dominant, and subdominant functions related by fifths, then the classical major triad on tonic is 0–1–8; on dominant, 8–9–16; and on subdominant, (-8)–(-7)–0.
 == Generalization ==